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Astronomy&Astrophysicsmanuscriptno.6222 c ESO2008 (cid:13) February5,2008 Magnetic fields in barred galaxies V. Modelling NGC 1365 D.Moss1,A.P.Snodin2,P.Englmaier3,A.Shukurov2,R.Beck4,andD.D.Sokoloff5 1 SchoolofMathematics,UniversityofManchester,OxfordRoad,Manchester,M139PL,UK 2 SchoolofMathematicsandStatistics,UniversityofNewcastle,NewcastleuponTyne,NE17RU,UK 7 3 InstituteofTheoreticalPhysics,UniversityofZu¨rich,Winterthurerstrasse190,8057Zu¨rich,Switzerland 0 4 Max-Planck-Institutfu¨rRadioastronomie,AufdemHu¨gel69,53121Bonn,Germany 0 5 DepartmentofPhysics,MoscowUniversity,119992Moscow,Russia 2 n Received...;accepted... a J ABSTRACT 0 Aims. WepresentamodeloftheglobalmagneticfieldinthebarredgalaxyNGC1365basedjointlyonthelarge-scalevelocityfield 1 ofinterstellargasfittedtoHandCO observationsofthisgalaxyandonmean-fielddynamotheory.Theaimofthepaperistopresent adetailedquantitativecomparisonofagalacticdynamomodelwithindependentradioobservations. 1 Methods. Weconsiderseveralgasdynamicalmodels,basedontworotationcurves.Wetestarangeofnonlineardynamomodelsthat v includeplausiblevariationsofthoseparametersthatarepoorlyknownfromobservations.Modelsforthecosmicraydistributionin 7 thegalaxyareintroducedinordertoproducesyntheticradiopolarizationmapsthatallowdirectcomparisonwiththoseobservedat 7 λλ3.5and6.2cm. 2 Results. Weshowthatthedynamomodelisrobustinthatthemostimportantmagneticfeaturesarecontrolledbytherelativelywell 1 established properties of thedensity distribution and gasvelocity field. The optimal agreement between thesynthetic polarization 0 mapsandobservationsisobtainedwhenauniformcosmicraydistributionisadopted.Thesemapsaresensitivetothenumberdensity 7 ofthermalionizedgasbecauseofFaradaydepolarizationeffects.Ourresultsarecompatiblewiththeobservedpolarizedradiointensity 0 andFaradayrotationmeasureifthedegreeofionizationisbetween0.01and0.2(withrespecttothetotalgasdensity,ratherthanto / h thediffusegasalone).Wefindsomeindirectevidenceforenhancedturbulenceintheregionsofstrongvelocityshear(spiralarmsand p large-scaleshocksinthebar)andwithin1–2kpcofthegalacticcentre.Weconfirmthatmagneticstressescandriveaninflowofgas - intotheinner1kpcofthegalaxyatarateofafewM yr−1. o Conclusions. Thedynamomodelsaresuccessfulto⊙someextentinmodellingthelargescaleregularmagneticfieldinthisgalaxy. r Ourresultsdemonstratethatdynamomodelsandsyntheticpolarizationmapscanprovideinformationaboutboththegasdynamical t s models and conditions in the interstellar medium. In particular, it seems that large-scale deviations from energy equipartition (or a pressurebalance)betweenlarge-scalemagneticfieldsandcosmicraysareunavoidable.Wedemonstratethatthedynamicaleffectsof : v magneticfieldscannotbeeverywhereignoredingalaxymodelling. i X Keywords.Galaxies:magneticfields–Galaxies:individual:NGC1365–Galaxies:spiral–ISM:magneticfields r a 1. Introduction featuresofthedynamotheorycertainlyarenotunderstoodwell enough. However, we demonstrate that gross features of the NGC 1365 is one of the best studied barred galaxies. It has modelgalacticmagneticfield–atleastinbarredgalaxieswhere been observed in a broad range of wavelengths, including H the shear in the large-scale velocity is the dominant induction (Ondrechen&vanderHulst1989),moleculargas(Curranetal. effect–areratherinsensitivetothepoorlyknowndetailsofthe 2001),Hα(Lindblad1999),andtheradiorange(Sanqvistetal. dynamosystem(mostimportantly,theα-coefficient).Therefore, 1995;Becketal.2005),inadditiontonumerousopticalandin- wecanplausiblyconstrainthefreedomwithinthedynamomod- fraredobservations(seeLindblad1999andreferencestherein). els, and so draw conclusions about the interstellar medium in DetailedgasdynamicalmodellingbyLindbladetal.(1996)pro- barredgalaxies. videdquantitativemodelsforthegravityandgasvelocityfields We findfairagreementbetweenradiopolarizationobserva- inthisgalaxythatfittheHand,tosomeextent,theCOobser- tionsandthemagneticfieldobtainedasasolutionofthemean- vations. field dynamo equations, using velocity and density fields ob- Theaimofthispaperistoaddtotheseeffortsbytheinclu- tained from gas dynamical simulations, although the distribu- sionofmagneticfields.Thegasdynamicalmodelofthegalaxy tion of polarized intensity is reproducedbetter than that of po- then can be tested against independentradio data, which were larization angles. Our models also support the idea that inter- notincludedintheconstructionofthemodel.Ofcourse,thisin- stellar turbulenceis enhancedin the vicinity of dust lanes near volves an additionalpiece of theory and some further assump- the bar major axis, and that the energy density of cosmic rays tions (concerning, e.g., the applicability of dynamo theory to can depend only weakly on position in the galaxy, thus devi- galaxies and uncertainties in some dynamo parameters). Some ating significantlyfromequipartitionwith interstellar magnetic field. As a result, radio polarizationobservationand modelling Sendoffprintrequeststo:D.Moss of magnetic fields are important ingredientsof both the theory 2 Mossetal.:ModellingNGC1365 andobservationsofbarredgalaxies.Thisworkresemblesquite kindlyprovidedtousbyP.O.Lindblad.Thispotential(the‘LLA strongly an earlier study of another barred galaxy, NGC 1097 model’inthefollowing)includesthegravitationalpotentialsof (Mosset al. 2001), butrepresentsa significantimprovementin the disc and spiral arms and was derived from the nonaxisym- that we now use a dynamical model that specifically models metric part of the deprojected J-band image. Their best fit pa- NGC1365,ratherthanthegenericdynamicalmodeladoptedfor rametersare A = 1.2and A = 0.3forthe relativecontri- bar spiral NGC 1097.Also, thedynamomodelwe use here isfullythree butionsofthebarandspiralarms.Themodelrotationcurvefits dimensional,whereasthatofMossetal.(2001)usedthe‘no-z’ theHrotationcurveforgalactocentricdistancesr > 120 and ′′ approximationtoremoveexplicitdependenceontheverticalco- givesreasonableresonancelocationsinsidethisradius.Various ordinate.Broadlycomparablestudieshavealso beenpublished versions of the LLA model used the bar angular velocity of by Otmianowska-Mazur et al. (2002), Soida et al. (2006) and Ω =18kms 1kpc 1(modelBSM)and17kms 1kpc 1(model p − − − − Vollmeretal.(2006). BSM2),withthecorotationradiuscloseto14kpcinbothcases. ThefullgravitationalpotentialoftheLLAmodelisobtained from two independent observations: (i) the H rotation curve, 2. Theobservedmagneticstructure usedtofixthetotalradialmassdistributionofthegalaxyinclud- NGC1365wasobservedintotalandpolarizedradiocontinuum ingdarkmatter,and(ii)theJ-banddata,tracingthestellarmass with theVLA DnCarrayatλ3.5cmandλ6.2cm. Thefullde- distribution,whichisonlyusedtoderive(afterdeprojection)the tails and the mapsat 15 and 25 angularresolutionare given amplitude of nonaxisymmetric perturbations in the disc plane. ′′ ′′ inBecketal.(2005).Thetotalradiointensity(ameasureofto- The latter cannot be used to derive the rotation curve reliably tal magnetic field strength and thermal emission) follows well becauseofthepresenceofdarkmatter,andtheformeralsocan the optical bar and the spiral arms. According to the observed bemisleadingwhenthegasflowissignificantlynonaxisymmet- spectralindices,thethermalfractionisabout20%atλ6.2cm. ric. Thepolarizedemission(Fig.1)isstrongestinthecentralre- Lindbladetal.(1996)adopted20Mpc(1 = 97pc)forthe ′′ gionandinnerbar,butdecreasesrapidlytowardsthe outerbar. distanceofNGC1365,butweadjustedthemodeltoadistance Thereisalsosignificantpolarizedemissionbetweenthebarand of18.6Mpc(1 =90pc)(Lindblad1999). ′′ the spiral arms. No concentrationin the spiral arms can be de- Isothermalgasdynamicalmodelswerecalculatedusingthe tected. At λ6.2cm, where the sensitivity is highest, the polar- codeZEUS2D,publishedbyStone& Norman(1992),andwe ized emission forms a smooth halo aroundthe bar. The degree found a close match to the model of Lindblad et al. (1996). ofpolarizationislowinthebarandspiralarms,indicatingthat However we did not attempt to take into account the warp in the turbulent magnetic field dominates in the regions of high the outer disc, as we are mostly interested in the inner region. gasdensityandstrongstar formation,while theregularfield is Our basic models, illustrated in Fig. 2b,c have the bar angu- strong between the bar and the spiral arms. At λ3.5cm, most lar velocity Ω = 16.16kms 1kpc 1 and the corotationradius oftheextendedpolarizedemissionoutsidethebarislostinthe p − − at R = 15.5kpc; we also considered a model (Fig. 2a) with noise because of the steep synchrotronspectrum. Furthermore, c Ω = 17kms 1kpc 1 and R = 16.3kpc. The angularvelocity thesensitivityoftheVLAtoextendedstructuresisreducedfor p − − c of the spiral pattern is taken to be equal to that of the bar. For scalesbeyond3arcminutesatλ3.5cm,whichaffectsthevisibil- reasonsexplainedbelowin Section 4, the resultinggas density ityofthelarge-scalepolarizedemissioninNGC1365,whileat in the bar region was too low to reproduce the observed mag- λ6.2cmthecriticallimitis5 arcminutesandso doesnotaffect neticfieldwithinthedynamomodel.ThegasdensityintheLLA ourobservations. modelcanbearguedtobeunderestimatedinsidethecorotation Thepeakpolarizedintensityis368mJyperbeamatλ3.5cm radiusbecausetherotationcurveusedhadpoorresolution,and inthemassivedustlanenortheastofthecentre(seeBecketal. underestimates the depth of the potential well. We derived our 2005). The fractional polarization is 0.8. At the same position basicmodelfromtheLLAmodel,byreplacingtherotationcurve the λ6.2cmmaprevealsa localminimumwith polarizedinten- usedbyLindbladetal.(1996)withthemorerecentCOrotation sityof150mJy/beam,correspondingtoafractionalpolarization curveofSofueetal.(1999).Thismodifiedmodelwasmuchbet- ofonly0.2,whichisneartheexpectedcontributionfrominstru- terabletoreproducetheobservedmagneticfield,whileremain- mentalpolarizationbythe brightnuclearregion.Thisindicates inginagreementwiththeoverallmorphologyofthemolecular thatstrongdepolarizationoccursatλ6.2cminthecentralregion, gasdistribution.Asignificantdifferenceisthatthereismorema- bya factorofatleast4.Inthe barandspiralarmsthedepolar- terialinthecentralregionswhenSofue’srotationcurveisused. izationfactoris2–3(Becketal.2005). TherotationcurveusedhereisshowninFig.3a,withtheposi- Polarizedemissioncanemergefromcoherent,regularmag- tionsofresonancesillustratedinFig.3b. netic fields or from anisotropic random magnetic fields; these possibilitiescanbedistinguishedwiththehelpofFaradayrota- We also studied the dependence of the gas dynamics and tionmeasures.InNGC1097,anisotropicfieldsdominateinthe magneticfieldonthesoundspeedadoptedintheisothermalgas barregion(Becket al. 2005).However,dueto the weak polar- model.Thisparameterisuncertaininourmodelsforseveralrea- ized intensity in NGC 1365, the observations available cannot sons. Englmaier & Gerhard (1997) showed that the large-scale providea large-scalemap of Faraday rotation,so that the rela- gasdistributionin isothermalgasflow modelsof barredgalax- tivecontributionsofcoherentandanisotropicrandommagnetic ies can depend on the sound speed, even if the pressure forces fieldsremainsunclear. arenegligible.Sincetheposition,andevenexistence,ofshocks dependsonMachnumber,theglobalgasflowconfigurationcan change as a result of a relatively small change in the speed of 3. Themodel sound. Different parts of the multi-phase interstellar medium (ISM) may not follow the same global gas flow. Different nu- 3.1.GasdynamicalmodelsofNGC1365 mericalmethodshavebeenshowntorepresentdifferentaspects WereproducedthegasdynamicalmodelofLindblad,Lindblad of the ISM with varying success. Sticky particle methods, for &Athanassoula(1996)usingtheirgravitationalpotential‘BSM’ example,modelbettertheclumpyISM,whilegrid-basedmeth- Mossetal.:ModellingNGC1365 3 Fig.1. The polarized intensity contours and magnetic vectors of the polarized radio emission at the wavelengths λ3.5cm (left hand panel) and λ6.2cm (right hand panel) (both smoothed to a resolution 25 ; the beam size is shown in the lower right of ′′ each panel), superimposedonto an ESO optical image of NGC 1365, kindly providedby P. O. Lindblad. The contour levels are 1,2,3,4,6,8,12,...times 30µJy/beam at λ3.5cm and 40µJy/beam at λ6.2cm; the r.m.s. noise is 15µJy/beam at λ3.5cm and 14µJy/beamatλ6.2cm. Fig.2. Themodelgasdensitywithsuperimposedvelocityvectorsinthereferenceframecorotatingwiththebar,ingasdynamical modelsbasedon(a)therotationcurveoftheLLAmodelwithc = 10kms 1 (lefthandpanel),andtherotationcurveofSofueet s − al.(1999)with(b)c =10kms 1(middlepanel)and(c)c =30kms 1(righthandpanel),withc thesoundspeed.Shadesofgrey s − s − s representthelogarithmofgasdensity(darkershadescorrespondingtolargervalues),witheachshadecorrespondingtothe same densityineachpanel.Notethesmallerdensitycontrastinthebarregioninthemodelwithhigherspeedofsound(panelc). ods give a better description of the shocks and the smooth gas Ourmagneticfield modelalso reliesonthegasdensityob- component. tainedfromgasdynamicalsimulationstogetherwiththevelocity field;thisisdiscussedinSect.3.2–seeEq.(4). TheglobalmagneticfielddependsonthegasflowviaEqs(1) and (2); however, it is not a priori clear which component of the ISM carries the magnetic field and, therefore, what is the appropriatesoundspeedofthegas.Wehaveconsideredmodels withthespeedofsoundequalto10and30kms 1(seeSect.5.4). − 4 Mossetal.:ModellingNGC1365 Table 1. Parameters of models discussed in the text, as defined in Sect. 3.2. In all the models, the angular speed of the bar is Ω =16.16kms 1kpc 1withthecorotationradiusat15.5kpc. p − − Model R η q r f c α 0 η η η s [1026cm2s 1] [kpc] [kms 1] − − 1 3.0 1.0 3 3.0 0 10 2 3.0 1.0 3 1.5 2 10 3 0.0 1.0 3 1.5 2 10 4 2.7 2.5 3 1.5 2 10 5 3.0 2.0 3 1.5 2 10 6 3.0 1.0 3 1.5 2 30 3.2.Thedynamomodel dimensionalgalacticdynamomodelsdescribedinMoss(1997), exceptthatcylindricalpolarcoordinateswereusedthere. Dynamo models, specifically simple mean-field turbulent dy- In Eq. (1), α parameterizesthe dynamoaction of the inter- namos, are remarkably successful in explaining the observed stellarturbulence,andηistheturbulentmagneticdiffusivity.We features of galactic magnetic fields (see, e.g., Ruzmaikin et al. assumebothofthesequantitiestobescalars(ratherthantensors) 1988; Beck et al. 1996; Widrow 2002 for reviews). Despite and,inordertoobtainasteadystatewithsaturateddynamoac- the fact that the nonlinear behaviour of turbulent dynamos is tion,introduceasimpleα-quenchingnonlinearityintotheprob- stillcontroversial,mean-fieldmodelsprovidearemarkablyreli- lem,writing ableempiricaldescriptionoflarge-scale(regular)galacticmag- neticfields(Shukurov2004).Fortunately,dynamosolutionsfor α= α0 , B2 =4πρ(r)v2 , (2) galaxiesarequiteinsensitivetothoseparametersthatarepoorly 1+ξB2/B2 eq t eq known, such as the form of the α-effect and even, to a lesser extent,theturbulentmagneticdiffusivity.Thisisespeciallytrue Ω(r) α =α f(z), (3) of models for barred galaxies where large-scale velocity shear 0 ∗ Ω0 plays a dominant role in determining magnetic field structure (Mossetal.1998a,2001);thentheprimaryroleoftheα-effect with istomaintainthefieldagainstdecay. sin(πz/h), z h/2, Ourmodelcanberegardedasadevelopmentofthedynamo | |≤ mneotidcelfioefldMionssaegteanl.e(r2ic00b1a)r,reudsegdatloaxmy.odWeelthneowlarignet-rsocdaulceemfaugr-- f(z)= cosh(2|z|/h−1)2 −1sgnz, |z|>h/2. therelaborationsrequiredtoreproducethebasicfeaturesofthe HereΩihsatypicalvalueoifΩ,B isthemagneticfieldstrength global magnetic pattern in NGC 1365. We solve the standard 0 eq correspondingto equipartitionbetween magnetic and turbulent mean field dynamo equation for the large-scale, regular mag- kineticenergies,andα isaconstant,whichwecanadjust.Quite neticfieldB arbitrarily,weadoptΩ∗ =Ωatr =3kpc,andEq.(3)showsthat 0 α is the maximum value of α at this radius. Thus we are as- ∂B = u B+αB 1 η B η B , (1) su∗mingthatthe large-scalemagneticfield significantlyreduces ∂t ∇×(cid:16) × − 2∇ × − ∇× (cid:17) the α-effectwhenits energydensityapproachesthatofthe tur- bulence;the constantξ is introducedto suggest formally some inthreespatialdimensions,usingCartesiancoordinates(x,y,z), of the uncertainty about the details of this feedback. The de- where xandyarehorizontaldimensions,andthediscmidplane pendenceofαonheight,definedby f(z),isimplicitlyoddwith isatz = 0.Hereαandηaretheturbulenttransportcoefficients respect to the midplane, with α increasing with z from 0 at responsiblefortheα-effectandturbulentmagneticdiffusion,re- | | | | z = 0toamaximumat z = h/2,andthendecreasingtozeroas spectively,u is the large-scalevelocityfield, and the term with | | z (rememberingthatweonlyexplicitlymodeltheregion η allows for the turbulent diamagnetism associated with the | | → ∞ ∇ z 0). Because of the symmetry of Eqs (1) and (2), if B is a spatialvariationof the turbulentdiffusivity(Roberts& Soward ≥ solution,then Bisalsoasolution. 1975). In our standard case, our computational domain covers − We take ξ = O(1), assuming that there is no catastrophic theregion L (x,y) L, 0 z aL = z , whereais the − ≤ ≤ ≤ ≤ max α-quenching(Brandenburg& Subramanian2005). The models domain’saspectratio.Wetakeameshofsizen n n ,with x× y× z werecomputedwithξ = 1,andthefieldstrengththenscalesas uniformspacinginthehorizontaldirectionsandalso,separately, ξ 1/2. The turbulent speed that enters B is taken to be equal vertically.The maximumresolutionreadilyavailable to uswas − eq to the speed of sound as adopted in the gas dynamical model. n =n =200,n =31,andinordertoresolvesatisfactorilythe x y z Thegasdensityρ(x,y,0)istakenfromthegasdynamicalmodel solutionswetook L = 15kpcanda = 0.12,so z = 1.8kpc. max describedinSect.3.1.Weextendthisawayfromz=0bywriting (Thuswestudyonlytheinnerpartofthisunusuallylargebarred galaxy.)Thetotalthicknessofgaslayerthathoststhelarge-scale ρ(x,y,0) magnetic field is taken as 2h = 0.9kpc, compatible with the ρ(x,y,z)= . (4) cosh(z/h) thicknessofthediffusewarmgasintheMilkyWay.Ourproce- | | dureistotime-stepthe xandycomponentsofEq.(1),andthen Themagnitudeofthegasdensityisrelativelyunimportantinour tousethecondition B=0toupdateB .Werestrictourselves model(wheretheLorentzforceisnotincludedintotheNavier– z ∇· tosolutionsofeven(quadrupolar)paritywithrespecttothedisc Stokesequation)asitaffectsonlythemagnitudeofthemagnetic planez = 0,andsothelatterstepisstraightforward,giventhat field in the steady state, via Eq.(2), butnotits spatial distribu- B = 0 at z = 0. This is the same procedureused in the three- tion.Theonlyaspectwherethemagnitudeofgasdensityplaysa z Mossetal.:ModellingNGC1365 5 code at attainable numerical resolution. Thus Ω was softened by introducing an explicit parabolic profile within a radius of 2.1kpc, with the maximumof Ω truncatedto 110kms 1kpc 1 − − (ascomparedto1730kms 1kpc 1 atr = 0.013kpc,thesmall- − − est distance from the axis in the gas dynamical model used). Thismodificationcan be expectedto reducethe magneticfield strengthinregionsclosetothegalacticcentre,butasthisregion isnotwellresolvedbytheradioobservations,wecannotinany casemakeacomparisonbetweentheseandthecomputedmag- neticfield. Further,wecontinuedthevelocityfieldabovethediscbyin- troducingz-dependenceintothehorizontalvelocitycomponents via u(x,y,0) u(x,y,z)= , (6) cosh(z/1.2kpc) | | andu =0everywhere. z Inordertomodela galaxysurroundedbynear-vacuum,we allow themagneticdiffusivityto becomelargehighin thehalo (Sokoloff&Shukurov1990), 1, z h, | |≤ η=η01+(η1−1)"1−exp −1|z.5|−kphc!#2 , |z|>h, whereη andη areconstants;thusη=η nearthediscmidplane 0 1 0 andη η η inthehaloregion(z >h).Weadoptedanominal 0 1 → | | η = 2 – larger values led to numerical difficulties. A conven- 1 tionalvalue of η is 1026cm2s 1; however,we also considered 0 − models with values larger than that – see Table 1. In order to reproduce polarized radio maps of NGC 1365 in sufficient de- tail,wehadtointroducefurtherspatialvariationinη.Following Moss et al. (2001), we have assumed that the turbulent diffu- Fig.3. (a): The rotation curves used in the paper: that from sivityisenhancedbytheshearofthenonaxisymmetricvelocity Lindbladetal.(1996) (solid;asinFig. 2a), andonemorecon- accordingto sistent with more recent CO observations (Sofue et al. 1999) (dashed; as in Fig. 2b,c). The plot assumes the distance of S ∂u ∂u η 1+ f , S = x + y , dNdiiGaugsCrao1mf3c6fo5orrotottahtbieoenr2o0itsaMtRiopcnc≈acsur1ivn4eskLpisnch.dob(wblan)d:ieTntha(eal.)li(nw1e9iat9rh6r)te.hsTeohnseaanmrcaee- w0h∝ere SmaxηSismtahxe!maximum v(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)a∂lyue(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) of(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)S∂x. T(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)he effect of fη , 0 line style. From bottom to top: Ω κ/2, Ω, and Ω + κ/2 in is, firstly, to broaden magnetic structures near the spiral arms, units of kms 1kpc 1. The resonan−ces are located at the in- and,secondly,toreducethecentralpeakofmagneticfield.The − − tersections with the horizontal lines corresponding to Ω = valuesof f adoptedareshowninTable1.Wedidnotconsider p η 16.16kms 1kpc 1 (solid) and 17kms 1kpc 1 (dashed). The asimilarenhancementinαasMossetal.(2001)foundittobe − − − − smallscalestructureintheΩ κ/2curvesisanartefactofplot- unimportant.Thevaluesof fη thatweresufficienttoproducere- ting. ± alisticmagneticfieldsinspiralarmwerestilltoosmalltoreduce the central maximum of magnetic field to an acceptable level. Therefore,weintroducedanadditionalenhancementofηinthe roleistheFaradaydepolarizationand,hence,themodelleddis- centralpartofthegalaxy,multiplyingη byq exp( r2/2r2);the tributionofpolarizedintensity.Thiseffectis,however,relatively 0 η − η valuesofq andr aregiveninTable1foreachmodelstudied. η η weakatλ = 3–6cmanditisplausiblethatotherdepolarization Clearly,wehavemadeanumberofratherarbitrarychoices, effects(e.g.,Faradaydispersion)aremoreimportantinthereal in particular when extending the two dimensional gas dynam- galaxy.ThegasdensityinourmodelisshowninFig.4. ical model into three dimensions. Our overallimpression from The gasvelocityin the plane z = 0, u(x,y,0),is also taken asubstantialnumberofnumericalexperimentsisthattheover- from the gas dynamical model. For convenience, we split this allnatureofourresultsdoesnotdependverystronglyonthese intorotationalandnon-circularparts, choices. Atz = z , andon x,y = L, theboundaryconditionsare u(x,y,0)=Ω(r)rφ+v(x,y,0), (5) max ± B = B = 0.Onz = 0,∂B /∂z = ∂B /∂z = 0, B = 0,andso x y x y z respectively,whebrer =(x2+y2)1/2isaxialdistance. theintegrationof B = 0givesthevaluesof Bz onthe other ∇· Wethenintroducedtwosignificantmodifications.Wefound boundaries. These are conservative boundary conditions on B x that, in the gas dynamical model, Ω(r) increases very rapidly and B , in that they will increase the field gradients and thus y towardsthe rotationaxis(veryapproximately,as 1/r).The gas raisethethresholdfordynamoactiontooccur. dynamicalmodelmodelappearstohandlethisfeaturesatisfacto- We nondimensionalize the problem in terms of the length rily,butitcausessignificantnumericalproblemsforthedynamo L = 15kpc, time h2/η and magnetic field B . Given that the 0 eq 6 Mossetal.:ModellingNGC1365 velocityfield,includingtheangularvelocity,isgivenbythedy- 0 5 10 16 21 namicalmodel,theonlyfreedynamoparameterisα ;thecorre- 15 spondingdimensionlessparameteris ∗ α h Rα = η∗ , (7) 10 0 whereα isdefinedinEq.(3).Thedynamoactionpreventsmag- netic fie∗ld from decay for values of R exceeding about 1 for α 5 η =1026cm2s 1;thecriticalvalueofR increasesroughlypro- 0 − α portionallytoη . 0 Henceforth,wewillusedimensionlessvariables,unlessex- c) p 0 plicitly otherwise stated; the units of gas number density and k ( magneticfieldstrengthare44cm 3and30µG,respectively. y − −5 4. Results Thegasdynamicanddynamomodelsdescribedabovetogether yieldthegasdensityandthedistributionofthelarge-scalemag- −10 netic field in the galaxy. The distribution of magnetic field in thegalaxyplaneresultingfromModel2(introducedinTable1), whichwearguebelowtobeourbestmodel,isshowninFig.4. −15 Giventhedistributionofthecosmicrays,wecannowconstruct −15 −10 −5 0 5 10 15 syntheticradioobservablesin ordertoassess thequalityof the x (kpc) model.We havecomputedsyntheticradiopolarizationmapsat wavelengthsof 3.5cm and 6.2cm using the dynamo generated Fig.4.Energydensitycontoursandvectorsoftheregularmag- magnetic field and the gas density, and compared them with netic field B from Model 2 (see Table 1), both at z = 0, are the observedradiomaps.Detailsof thisprocedurearegivenin showntogetherwithgasnumberdensityrepresentedwithshades AppendixA. Since we do not modelturbulentmagnetic fields, of grey. The contours shown correspond to approximately 0.1, weareunabletocalculatethetotalradiointensityandtoestimate 0.6 and 3.0 times the r.m.s. value; the length of the vectors is thedegreeofpolarizationfromthemodel. proportionaltoB2.Thescalebaratthetopoftheframerefersto Weconsideredseveralmodelsforthenumberdensityofcos- thegasnumberdensityintheunitsofhydrogenatomspercm3. micrays,n ,whichwediscussinSect.5.1.Forallbutoneofthe cr dynamomodelslistedinTable1wefindthatthesimplestpossi- blechoice,n =const,providesthebestfittotheobserveddata, cr Weusedtwomaintechniquestocomparethesyntheticmaps regardlessoftheotherquantitiesadopted.Thelargervalueofr η with observations and therefore to select the optimal magnetic in Model 1 producesa relatively weak magnetic field through- field model. We chose to use the λ6.2cm map of polarized in- outalargecentralregioncomparedwiththatattheendsofthe tensity in the analysis since it has the best signal-to-noise ra- bar. In order to fit the observed central peaks of polarized in- tio. All model data, including synthetic radio maps, have been tensity,P,animplausiblenon-uniformdistributionofn would cr smoothed(intermsoftheStokesparametersQandU)tomatch berequiredin thismodel.Specifically,thecosmic raydistribu- the resolution of the observations.In Sect. 4.2 we comparethe tion required to reconcile this model with observations would distributions of polarized intensity on cuts along various paths haveahighpeakwithin3kpcofthecentrewheremagneticfield in the plane of the sky. In Sect. 4.3 we analyse the difference strengthisminimum.Fortheotherdynamomodels,anyplausi- betweenthecomputedandobservedpolarizedintensitiesintwo blenon-uniformdistributionofn producestoostrongacentral cr dimensions.Inaddition,wecomparetheorientationsofthemag- maximumof P relativetoall otherstructures.Inparticular,the neticB-vectorsobtainedfromtheobservedandsyntheticStokes polarized intensity in the spiral arms is almost lost in models parameters(Sect.4.5). withnon-uniformn ,beingfarweakerthanthatwithin1–2kpc cr of the centre. Since we have truncated the angular velocity at To rotate the modelgalaxyto the positionof NGC 1365in r < 2.1kpc, the untruncateddifferentialrotation would lead to the sky, we took the inclinationangle i = 46◦ and the position anevenstrongerdiscrepancy. angle of the galaxy’smajor axis (i.e the intersectionof the sky Oursyntheticmapsdonotincludeanydepolarizationeffects planeandthegalaxyplane)PA=222◦,whicharethoseassumed due to random magnetic fields (see Burn 1966; Sokoloff et al. inobtainingtherotationcurveforour(favoured)gasdynamical 1998),althoughtheyallowfullyfordepolarizationbytheregu- model.Resultsarequitesensitivetothesevalues,anditispos- larmagneticfields(differentialFaradayrotationandbeamdepo- siblethatareappraisalcouldresultinnoticeablechanges. larization).InordertoincludeFaradaydepolarizationeffectsdue tothelarge-scalemagneticfield,weassumedanominalconstant 4.1.Syntheticpolarizationmaps ionizationfractionof X = n /n = 0.1,correspondingto ather- e malelectrondensityof0.1ofthetotalgasdensityobtainedfrom Overall, Model 2 (specified in Table 1) appears to provide the thegasdynamicalsimulationsasdescribedinSect.3.2.Guided bestfittotheobservedpolarizationmap;Model4isonlyslightly byanalogywiththeMilkyWay,wheretheaveragetotalgasden- worse – see Sect. 4.2. Contoursof B2 shown in Fig. 4 indicate sityis1cm 3whereasthethermalelectrondensityis0.03cm 3, thattheregularmagneticfieldisstrongerinthebarregionwhere − − asmallervalueofXmightbeappropriate.WeshowresultsforX gasdensityislarge,andoutsidetheregionsofhighdensityinthe closetothisvalueinSect.4.1.InSect.5.5,wediscusstheeffect spiralarms.Therearemagneticfeaturesapparentlyunrelatedto ofvariationsinX andarguethat0.01< X <0.2. thedensitydistribution[e.g.,thosepassingthroughthepositions ∼ ∼ Mossetal.:ModellingNGC1365 7 150 200 (a) 150 100 P 100 50 50 ) c e cs 0 0 r a ( y −100 0 100 distance from centre (arcsec) −50 300 (b) 250 −100 200 P 150 −150 −150 −100 −50 0 50 100 150 100 x (arcsec) 50 Fig.5. Asyntheticmapofpolarizedsynchrotronintensity(con- 0 tours)andpolarizationplanesatλ6.2cm,resultingfromModel2 −100 0 100 (seeTable1)assumingthatn =const,areshownsuperimposed cr distance from centre (arcsec) on the optical image of the galaxy NGC 1365 (shown in only a few shades of grey for clarity). The synthetic map has been 300 (c) smoothedtotheresolutionof25 tomatchthatoftheobserved ′′ 250 map shown in Fig. 1. The contour levels shown are approximately (1,3,6,12,32) P /45, where P is the maximum max max 200 × ofPinthesyntheticmap.Dashedlinesshowthepositionofcuts discussedinSect.4.2. P 150 100 (x,y) ( 5,8),(5, 8)];they are presumably formed by a lo- 50 ≈ − − cally enhanced velocity shear. The magnetic field has a deep minimum within the bar, mainly producedby the density defi- 0 ciencyinthatregion.Otherimportantfeaturesclearlyvisiblein −100 0 100 Fig. 4 are the magnetic field enhancementsin the dustlane re- distance from centre (arcsec) gion,wheremagneticfieldisamplifiedbybothcompressionand Fig.6. Cuts, at position angle 31 passing through the galac- ◦ shear,andtheprominentcentralpeak. − tic centre (left to right in the plots corresponds moving from The synthetic polarization map for this model is shown in south-easttonorth-westinthesky),throughpolarizedintensity Fig.5.Thiscanbecompareddirectlywiththeobservedmapin mapsatλ6.2cmsmoothedtoHPBW=25 ,for(a)theobserved ′′ theright-hand-panelofFig.1;themaps(andallothermapswe map,andsyntheticmapsfrom(b)Model2and(c)Model4,both show)areatasimilarscaletofacilitatethecomparison;wemake for n = const. In panels(b) and (c), the synthetic profiles for cr this comparison more quantitatively in Sect. 4.3. Our models λ6.2cm and λ3.5cm are shown solid and dotted, respectively; havea highdegreeofsymmetry,whereasthe ‘real’NGC1365 thedifferenceisduetoFaradayandbeamdepolarizationforthe isonlyapproximatelysymmetric;sincetheobservedmaplooks assumed ionization degree X = 0.1. The units of P are as in more regular on the eastern side, we shall mostly refer to that Fig. 1 for (a) and arbitrary in (b) and (c), but adjusted to fit a part of the galaxy unless stated otherwise. Despite the differ- similarrange.Thedottedprofilesforλ3.5cmwithX = 0.1also ence in symmetry,there is broadagreementbetweenthese two correspondtoPatλ6.2cmwithX =0.032. maps; for example, both have a deep minimum of P near the bar’s major axis where gas density is low, and both have the magneticspiralarmsdisplacedfromthegaseousones(although bothmagneticarmsaredisplacedtolargerradiiinthesynthetic ical model underestimates significantly the amount of molecu- map, only one arm is so displaced in the observed map). The lar gas in the bar region. Synthetic P is large both to the north minimumofthesynthetic Pinthebar(correspondingalsotoa and southof the bar majoraxis. In particular,the modelrepro- minimum of magnetic field within the bar, as seen in Fig. 4), ducesamaximumof Pupstreamofthebarmajoraxis,centred is broader than of the observations (see Sect. 4.2). The rea- intheλ6.2cmmapofFig.1at(RA=03h33min40sec,Dec= son for this is the very low gas density in this region, lead- 36 0900 ). These maxima apparentlyarise fromslightly en- ◦ ′ ′′ − ing to weaker magnetic fields via Eq. (2). This feature is fur- hanced velocity shear (that locally amplifies magnetic field) ther discussed in Sect. 6 where we argue that the gas dynam- rather than from local density maxima. We also note maxima 8 Mossetal.:ModellingNGC1365 200 (a) 200 (a) 150 150 100 P P 100 50 50 0 0 −150 −100 −50 0 50 100 150 −150 −100 −50 0 50 100 150 distance from centre (arcsec) distance from centre (arcsec) 250 (b) (b) 250 200 200 150 150 P P 100 100 50 50 −150 −100 −50 0 50 100 150 −150 −100 −50 0 50 100 150 distance from centre (arcsec) distance from centre (arcsec) (c) (c) 250 200 200 150 P 150 P 100 100 50 50 −150 −100 −50 0 50 100 150 −150 −100 −50 0 50 100 150 distance from centre (arcsec) distance from centre (arcsec) Fig.7.AsinFig.6,butatpositionangle0 (lefttorightissouth Fig.8. As in Fig. 6, but at position angle 90 (left to right is ◦ ◦ − tonorthinthesky). easttowestinthesky). of P near the ends of the bar and the beginning of the spiral length(seeSect.2)andhastobetakenintoaccountwhencom- arms, at (RA = 03h33 min45sec,Dec = 36 0815 ) and paring the model and observations. We use cuts through the ◦ ′ ′′ − (RA=03h33min28sec,Dec= 36 0830 ).Wenotethatthe centre of the galaxy at position angles PA = 0 , 90 and − ◦ ′ ′′ ◦ − ◦ observedtotalemission (notshownhere;see Beck etal. 2005) 31 , where PA is measured counterclockwise from the north ◦ − isrelatedtogasdensityinaratherstraightforwardmannerbeing as shown in Fig. 5. (The angle 31 is chosen so that the cut ◦ − correlatedwiththegasdensity.Thefactthatthisisnotthecase goes throughthe spiral arms; this correspondsroughlyto a di- with the polarizedintensity (as seen in both observedand syn- agonal in the computationalframe of Fig. 4.) The positions of theticmaps)confirmsthattheobservedregularmagneticfieldis these cuts are shown in Fig. 5. The synthetic P has been nor- notfrozenintothegas,apparentlybeingaffectedbythedynamo malized to make the mean difference between that and the ob- action. served P approximately zero. We have superimposed another profilefromcutsthroughsyntheticmapswhichrepresentsboth P at λ3.5cm with our favoured value of X = 0.1, and also 4.2.Cutsthroughpolarizationmaps at λ6.2cm with X = 0.032 (incidentally, this is close to the mean ionization degree of the warm diffuse gas in the Milky We found that comparisons can be usefully quantified and de- tailedusingthecutsintheskyplanementionedabove.Weshow Way). The coincidence of these two cuts is due to equivalent cuts only through the map at λ6.2cm because this map has Faradaydepolarization,whichdependsdirectlyon thequantity higher signal-to-noise ratio and includes the large-scale emis- ψ(z) λ2X ∞nB dzalongthelineofsighttowardsanobserver ∝ z k sion fully. However, depolarization is significant at this wave- (see AppendRixA). Here B is the line of sightfield component k Mossetal.:ModellingNGC1365 9 and n = Xn, therefore λ2X = const identifies equivalence in correspondinglyhigherspeedofsound,whichwouldallow the e depolarization.We seethatthisvalueisaboutthesameinboth turbulentspeedtobelargerthanelsewhere. cases(i.e.6.22 0.032 3.52 0.1).Thedifferencebetweenthe TheModel2cutatPA= 31 (Fig.6),whichpassesthrough ◦ × ≈ × − polarizationforthesetwopossibilitiesisthenjustaλ-dependent thespiralarms,showsanencouragingagreementwithobserva- scalefactor. tions.Forexample,Bhasamaximumslightlyoutsidethenorth- The cuts are presented in Figs 6, 7 and 8 for the best-fit ernarminboththismodelandtherealgalaxy.However,theout- Model 2 and also for Model 4. The latter model has the back- ermostmaximaproducedbythespiralarmsareslightlytoofar ground turbulent magnetic diffusivity η enhanced by a factor awayfromthecentreinthemodel.AsillustratedinFig.6(b),the 0 of 2.Thisleadsto a significantlysmoother,less structureddis- relativeheightsofthepeaksatλ6.2cmaresignificantlyaffected tribution of P. Thus, comparison of Models 2 and 4 allows us byFaradayrotationevenfor X = 0.1,wheretheyclearlydiffer to suggest that the effective turbulent magnetic diffusivity in bymorethanjustascalefactorbetweenλ6.2cmandλ3.5cm. the interstellar gas of barred galaxies is, on average, close to ThecutatPA = 0(Fig.7)exhibitssimilardegreeofagree- η = 1026cm2s 1. This value is typical of spiral galaxies in ment with the observations. The main deficiency of the model 0 − generalandis thatobtainedif the turbulentspeed v is close to here is the too narrowdistributionof P (the magneticstructure t 10kms 1andtheturbulentscaleisaboutl=0.1kpc;η 1lv. ofthemodelistoopooroutsidethebar)andtheminimumistoo − 0 ≃ 3 t Ourmodelneglectsdepolarizationdueto randommagnetic deepnearthecentreofthecut. fieldswhichcanreducethevalueofPinthecentralpartsmore ThecutatPA= 90 inthesyntheticmap,showninFig.8, ◦ − stronglythanintheoutergalaxyandthereforeaffecttherelative hasacentralmaximumthatistoonarrow(oroff-centreminima height of the central peaks in Figs 6–8. Depolarization due to that are too broad). This difference results in the deep minima internalFaradaydispersionreducesthedegreeofpolarizationto in the difference parameter δ discussed in Sect. 4.3. The sharp minimumintheobservedcutnearthecentreisaresultofbeam 1 e S depolarization;it occursin thesyntheticcutsas well, butisre- p= p − − , (8) 0 S movedbysmoothing. Model 2 seems to be almost optimal. The model could whereS = 2σ2 λ4 withσ2 = 2C2 b2 n2 dLthevarianceof be fine tuned by changing η and r within the ranges (1– RM RM 1h ih ei 0 η the Faraday rotation measure. HereC1 is the dimensionalcon- 2) 1026cm2s−1 and1.5–3,respectively.Forexample,thesec- stant appearing in the definition of the Faraday rotation mea- ond×arypeaksinthePA=0cutdecreaseinstrengthinModel4. sure (see AppendixA), b is the turbulentmagneticfield, angu- Further,increasingn byafactorof2withinthecentral1.5kpc cr larbracketsdenoteaveraging(thefluctuationsinmagneticfield wouldmakethecentralpeakhigher.However,wehavenotmade and thermal electron density are assumed to be uncorrelated), suchposthocadjustments. d is the turbulentscale and L is the pathlength(Sokoloffet al. 1998).Thebestavailableestimateoftherandommagneticfield in the central region of NGC 1365, b 40µG, follows from 4.3.Thedifferencemaps ≃ the total synchrotron intensity assuming equipartition between To obtaina globalcomparisonofthe modelsandobservations, cosmicraysandmagneticfields(seehoweverSect.5.1foradis- we producedmaps of the difference between the observed and cussionofthevalidityofthisassumption).Forn = 0.03cm 3, e − synthetic polarization at λ6.2cm, with the synthetic polariza- d = 0.1kpc, L = 1kpcandλ = 6.2cm, wethenobtainS 6, tionscaledtomakethemeandifferenceapproximatelyzero;this ≃ implying that this mechanism can depolarize the central peak measurewasfurthernormalizedbydividingthedifferencebythe significantly, giving p/p 0.2. Since the height of the sec- 0 appropriatelynormalizednoiseleveloftheobservedmapgiving ≃ ondarypeakshouldalsobeaffectedbydepolarization,albeitto a lesser extent, we expect that the ratio of the two peaks will (1.4P/P ) (P/P ) bereducedbyafactorsmallerthanfive.Wenote,however,that δ= max model− max obs . (9) (σ /P ) P max obs this estimate is uncertain since the number density of thermal electrons, their filling factor, turbulentscale and other parame- Thus, all comparisonswere performedpointwise after their re- ters are not knownwell enough.An alternative is to assess the ductiontothecommonresolution25 –thisisquiteastringent ′′ importanceofthisdepolarizationeffectbycomparingpolarized testofthemodel.TheresultisshowninFig.9forModel2. intensitiesatλ6.2cmandλ3.5cm.Theratioofthecentralpeak Since the models– unlikethe realgalaxy– possess perfect tothesecondaryonesatλ3.5cmisabout6–8,asopposedto2–3 symmetry,the differencecan hardlybe uniformlysmall: a per- atλ6.2cm.ThedifferencecanbeattributedtoFaradaydepolar- fectfitinonehalfofthegalaxywouldproducesignificantsys- ization(bybothregularandrandommagneticfields).Assuming tematic discrepancyin the other half. With this caveat, the dif- thatFaradaydepolarizationatλ3.5cmisnegligible,weconclude ferencemapshowsanacceptableglobalagreementofthemodel thatitcanreducethedegreeofpolarizationatλ6.2cmbyafac- with observations, in that it does not show much of the basic toraslargeas4,whichisconsistentwiththeanalyticalestimate. morphologicalelements of the galaxy.The normalized relative We conclude that the relative height of the central peak in the differenceisabout6–14infourspotsobservedtotheeast,south syntheticcutsofFigs6–8wouldbereducedbyFaradaydisper- and north-west of the galactic centre, indicating that synthetic sion,althoughthisisdifficulttoestimateaccurately. polarized intensity is too small upstream of the dust lanes and Given the above uncertainties in the amountof depolariza- at two positions at the inner edge of the western spiral arm. tion,allthreecutsforModel2aresimilartothoseobserved.In Otherwise, δ < 4 across the whole field of view. Given the particular, the relative heights of the peaks in P and, more im- limitedscop|e|o∼fourmodel(e.g.,itdoesnotincludeanyturbu- portantly,thepositionsofbothmaximaandminimaareremark- lentmagneticfieldswhichcanproducepolarizedradioemission ably realistic. The characteristic feature of this model is that η where they are anisotropic), we consider this degree of agree- is further enhanced by a factor of q 3 in the inner region menttobeacceptable.WediscussinSect.5.1acosmicraydis- η of NGC 1365, r < 3kpc. This enhanc≃ement can be due to a tribution that would provide perfect fit of Model 2 to observa- higher rate of star∼formation, and hence more hot gas, with a tions. 10 Mossetal.:ModellingNGC1365 150 100 50 ) c e s c 0 r a ( y −50 −100 −150 −150 −100 −50 0 50 100 150 x (arcsec) Fig.9.Thedifferenceδbetweennormalizedsyntheticandobservedpolarizationmapsatλ6.2cm, asdefinedin Eq.(9),superim- posedontheopticalimageofNGC1365.Thecontourspacingis2,withthezerocontourshownsolid,negativevaluesofδdashed, andpositive,dotted. 4.4.Faradayrotation so that modes with odd values of m do not occur in the mod- elled magnetic field. The contribution of the m = 1 mode to Wecanusepolarizedintensity(asinthecomparisonsabove)to those Fourier expansions is more important than just produc- probethedistributionofthelarge-scalemagneticfieldstrength, ing the overall asymmetry. In particular, superposition of vari- and also to deduce the orientation of the magnetic field in the ous azimuthalmodes produceslocal magnetic features at kilo- plane of the sky (via polarization vectors). However, knowl- parsec scale which are lost if only even modes are retained in edgeofthisquantitydoesnotdeterminethefielddirection.The the observedstructure to facilitate comparison with the model. FaradayrotationmeasureRMissensitivetothedirectionofthe Therefore,wedidnotfinditusefultocomparethemodelledand magneticfield,buttheobservedRMmapisverypatchybecause observed magnetic structures in this manner. (The presence of ofthelowersignal-to-noiseratioatλ3.5cm.Therefore,weused unmodelledodd-m structure was also a feature of our study of RMdataonlytoestablishaminimumacceptabledegreeofgas NGC1097inMossetal.2001.) ionization. Weinsteadcomparedirectlytheorientationofthemagnetic field vectors in the observed and synthetic polarization maps. Comparison of two-dimensional vector fields is difficult. We 4.5.Magneticfieldstructure couldapproachthisbytakingcutsthroughmapsofthemagnetic An analysis of the observed global magnetic structure in fieldorientationangles,aswasdonewiththepolarizedintensity. NGC1365thatissensitivetothedirectionofmagneticfieldwas However,asmallshiftinafeaturesuchasashockfrontcanre- performedbyBecketal.(2005)byfittingthepolarizationangles sult in drastic differences between any such cuts made parallel obtainedfrommulti-frequencyobservations.Thisanalysispro- tothefront. videsthelarge-scalemagneticfieldexpandedintoFourierseries InFig.10weshowtheorientationofboththesyntheticand in the azimuthal angle. Their results indicate the presence of a observedmagneticfieldvectorsobtaineddirectlyfromthecor- significantcomponentwiththeazimuthalwavenumberm=1at responding Stokes parameters; points below 3 times the r.m.s. almostall distancesfrom the galactic centre.However,our un- noise levelare neglected in the observedmaps. Agreementbe- derlying gas dynamical model has even symmetry in azimuth, tween modeland observationsis reasonable in the top left and